Variational approach to solutions for a class of fractional boundary value problem

摘要

In this paper we investigate the existence of infinitely many solutions for the following fractional boundary value problem egin{equation*} egin{cases} _tD^{lpha}_T(_0D^{lpha}_t u(t))=abla W(t,u(t)),qquad tin [0,T], u(0)=u(T)=0, end{cases} ag{FBVP} end{equation*} where $lphain (1/2,1)$, $uin mathbb{R}^n$, $Win C^1([0,T]imesmathbb{R}^n,mathbb{R})$ and $abla W(t,u)$ is the gradient of $W(t,u)$ at $u$. The novelty of this paper is that, assuming $W(t,u)$ is of subquadratic growth as $|u|ightarrow+infty$, we show that (FBVP) possesses infinitely many solutions via the genus properties in the critical theory. Recent results in the literature are generalized and significantly improved.